Method for phase retrieval to reduce a sampling requirement when imaging a dynamic process

ABSTRACT

A method for retrieving phase information in a coherent diffraction imaging process includes acquiring a plurality of 3D data sets, each 3D data set corresponding to one of a plurality of time states, and reconstructing a 3D image of the object at a given time state using the 3D data set from all of the time states. Each 3D data set is acquired by: illuminating an object positioned in a first position with a coherent beam; measuring a first 2D diffraction pattern using an area detector; rotating the object around a tilt axis thereof to a second position that is different from the first position; re-illuminating the object positioned in the second position with the coherent beam; re-measuring a second 2D diffraction pattern using the area detector; and repeating the rotating, re-illuminating and re-measuring steps such that each 3D data set includes a predetermined number of diffraction patterns.

STATEMENT OF GOVERNMENT INTEREST

The United States Government claims certain rights in this inventionpursuant to Contract No. DE-AC02-06CH11357 between the U.S. Departmentof Energy and UChicago Argonne, LLC, as operator of Argonne NationalLaboratories.

FIELD OF THE INVENTION

The present invention relates generally to the field of imagingtechniques that do not produce an image directly. More specifically, thepresent invention relates to a method for phase retrieval in an imagingtechnique, in particular, to phase retrieval during a dynamic process.

BACKGROUND

This section is intended to provide a background or context to theinvention recited in the claims. The description herein may includeconcepts that could be pursued, but are not necessarily ones that havebeen previously conceived or pursued. Therefore, unless otherwiseindicated herein, what is described in this section is not prior art tothe description and claims in this application and is not admitted to beprior art by inclusion in this section.

Understanding nanoscale processes is key to improving the performance ofadvanced technologies, such as batteries, catalysts, and fuel cells.However, many processes occur inside devices at short length and timescales in reactive environments and represent a significant imagingchallenge. One way to study such structures is by using coherentdiffraction imaging (CDI). CDI is a lensless technique forreconstructing a 3D image of an object based on a diffraction pattern.In CDI, a coherent beam of x-rays or electrons is incident on an object,the beam scattered by the object produces a diffraction pattern that ismeasured by an area detector, and an image is reconstructed from thediffraction pattern. When the diffraction pattern is measured by thearea detector, the data is based on a number of counts of photons orelectrons. This gives rise to the “phase problem” or loss of phaseinformation when a diffraction pattern is collected. In order to recoverthe phase information, Fourier-based iterative phase retrievalalgorithms are utilized.

Another example of CDI is Bragg coherent diffractive imaging (BCDI). InBCDI, the object is rotated around a tilt axis and a sequence of 2Ddiffraction patterns are measured at different sample orientations. BCDIis a powerful technique for investigating dynamic nanoscale processes innanoparticles immersed in reactive, realistic environments. With currentBCDI methods, 3D image reconstructions of nanoscale crystals have beenused to identify and track dislocations image cathode lattice strainduring battery operation, indicate the presence of surface adsorbates,and reveal twin domains. The temporal resolution of current BCDIexperiments, however, is limited by the oversampling requirements forcurrent phase retrieval algorithms.

Conventional phase retrieval algorithms present a problem in improvingthe time resolution of an imaging technique (e.g., CDI, BCDI, etc.). Theimaging technique requires a computer algorithm to convert theexperimental data into a real space image of the object (e.g.,nanocrystals) in reactive environments. Oversampling refers to thenumber of measurements that are required to faithfully reproduce theimage. The autocorrelation of the image must be sampled at the Nyquistfrequency, which is twice the highest frequency in the system. Becauseeach measurement takes time, the oversampling requirement results in alengthy process that may include prolonged radiation dose. Moreover, ifthe oversampling requirement is too high, it may not be possible toimage a dynamic process such as crystal growth, catalysis or strain inbattery cathode nanoparticles that occur too rapidly (i.e., faster thanthe amount of time it takes to satisfy the oversampling requirements).

A need exists for improved technology, including a method for phaseretrieval that allows for the time resolution to be improved, forexample, by reducing the oversampling requirement at each time step.

SUMMARY

One embodiment of the invention relates to a method for retrieving phaseinformation in a coherent diffraction imaging process. The methodincludes acquiring a plurality of 3D data sets, each 3D data setcorresponding to one of a plurality of time states and reconstructing a3D image of the object at a given time state using information from the3D data set from the given time state and all of the other time states.Each 3D data set is acquired using the following steps: illuminating anobject to be imaged with a coherent beam, the object positioned in afirst position; measuring a first 2D diffraction pattern using an areadetector that detects a number of photons or electrons in a beamscattered by the object; rotating the object around a tilt axis thereofto a second position that is different from the first position;re-illuminating the object with the coherent beam, the object positionedin the second position; re-measuring a second 2D diffraction patternusing the area detector; and repeating the rotating, re-illuminating andre-measuring steps a predetermined number of times such that each 3Ddata set includes a predetermined number of diffraction patterns.

In some aspects, the predetermined number of diffraction patterns in atleast one 3D data set is less than a minimum number of 2D diffractionpatterns required based on the Nyquist frequency. The 3D imagereconstructed based on the at least one 3D data set has image fidelity,where image fidelity is established when a sum of a difference between areal image and the 3D image reconstructed based on the at least one 3Ddata set on a pixel by pixel basis is less than or equal to apredetermined level of acceptable error. The predetermined level ofacceptable error may be, for example, less than or equal to 10% perpixel.

In some aspects, the predetermined number of diffraction patterns in atleast one 3D data set is ⅓ of a minimum number of 2D diffractionpatterns required based on the Nyquist frequency.

In some aspects, the plurality of 3D data sets include a 3D data setacquired at an initial time state, a 3D data set acquired at a finaltime state, and at least one 3D data set acquired at an intermediatetime state between the initial time state and the final time state.

In some aspects, the predetermined number of diffraction patterns in theinitial time state is equal to a minimum number of 2D diffractionpatterns required based on the Nyquist frequency. The predeterminednumber of diffraction patterns in the final time state is equal to theminimum number of 2D diffraction patterns required based on the Nyquistfrequency. The predetermined number of diffraction patterns in theintermediate time state is less than to the minimum number of 2Ddiffraction patterns required based on the Nyquist frequency.

In some aspects, the object is a nanocrystal having a size, for example,of 100-500 nm. The step of reconstructing the 3D image of thenanocrystal is repeated for each of the plurality of time states toproduce a series of 3D images that show a change in a structure of thenanocrystal over time.

Additional features, advantages, and embodiments of the presentdisclosure may be set forth from consideration of the following detaileddescription, drawings, and claims. Moreover, it is to be understood thatboth the foregoing summary of the present disclosure and the followingdetailed description are exemplary and intended to provide furtherexplanation without further limiting the scope of the present disclosureclaimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The disclosure will become more fully understood from the followingdetailed description, taken in conjunction with the accompanyingfigures, in which:

FIG. 1(A) illustrates a Bragg coherent diffractive imaging (BCDI)experiment of a single time-evolving nanocrystal. In particular, FIG.1(A) is a schematic illustration of a Bragg rocking curve that sweepsthe 2D area detector through the 3D volume in the q_(z) direction.

FIG. 1(B) illustrates a Bragg coherent diffractive imaging (BCDI)experiment of a single time-evolving nanocrystal. In particular, FIG.1(B) illustrates select time states from a time sequence that wasreconstructed from experimental BCDI data measured with greater than therequired oversampling.

FIG. 2 illustrates a nearest-neighbor correlation coefficient of u₁₁₁(r)for all pairs in the chosen time sequence. The average nearest-neighborcorrelation coefficient is 61%.

FIG. 3(A) illustrates a time correlation in the [111] displacement fieldprojection u₁₁₁(r) during the reconstructed time sequence.

FIG. 3(B) illustrates an average (over all time states and randomstarts) modulus error and average (over all time states and randomstarts) miscorrelation term normalized by the total intensity in theimage as a function of the scalar weight. Error bars represent thestandard deviation of the average over all time states obtained from tendifferent random starts.

FIG. 4(A) illustrates the results for reduced oversampling. Inparticular, FIG. 4(A) illustrates the average (over all time states andrandom starts) normalized modulus error as a function of the scalarweight w for ⅓, 1/20, and 1/84 of the required oversampling. The moduluserror reported here is computed with respect to the data with therequired oversampling. Error bars represent the standard deviation ofthe different average (over all time states) values obtained from tendifferent random starts. The error bar for w=0.01 and 1/20th of therequired oversampling is offset for clarity. The black dashed horizontalline shows the normalized average modulus error using the average of theinitial and final time states for every time state in the sequence.

FIG. 4(B) illustrates the results for reduced oversampling. Inparticular, FIG. 4(B) illustrates the individual time state errors for ⅓required oversampling for a particular random start for w=0 (opencircle), w=0.01 (points), and w=0.5 (×).

FIG. 5 illustrates three cross sections of the real data from t=12 minused to test the chrono CDI algorithm. The scale is the log₁₀ of thenumber of photons. The data is oversampled by a factor of 3 in q_(z) andhas both noise and a finite scattering extent. These measured data sets,after background subtraction (1-2 photons) and the removal of a numberof 2D slices, are used to test the chrono CDI algorithm.

FIG. 6(A) illustrates reconstructions of experimental measurement datawith reduced oversampling. The real part of the image (shown as anisosurface) corresponds to the reconstructed Bragg electron density,while the complex part of the image (gradient map projected onto theisosurface) corresponds to the reconstructed displacement fieldprojection. In particular, FIG. 6(A) illustrates the t=12 minreconstructions for ½ of the required oversampling for w=0 and w=0.01,and the true solution, which is reconstructed using the complete dataset.

FIG. 6(B) illustrates reconstructions of experimental measurement datawith reduced oversampling. The real part of the image (shown as anisosurface) corresponds to the reconstructed Bragg electron density,while the complex part of the image (gradient map projected onto theisosurface) corresponds to the reconstructed displacement fieldprojection. In particular, FIG. 6(B) illustrates the t=42 minreconstructions for ½ of the required oversampling for w=0 and w=0.01,and the true solution, which is reconstructed using the complete dataset.

FIG. 6(C) illustrates reconstructions of experimental measurement datawith reduced oversampling. The real part of the image (shown as anisosurface) corresponds to the reconstructed Bragg electron density,while the complex part of the image (gradient map projected onto theisosurface) corresponds to the reconstructed displacement fieldprojection. In particular, FIG. 6(C) illustrates the t=66 minreconstructions for ½ of the required oversampling for w=0 and w=0.01,and the true solution, which is reconstructed using the complete dataset.

FIG. 7 illustrates reconstructions of real measurement data with 3/10 ofthe required oversampling and w=0.01. The real part of the image (shownas an isosurface) corresponds to the reconstructed Bragg electrondensity, while the complex part of the image, φ, is proportional to thedisplacement field projection. Three states from the reconstructed timesequence are shown.

FIG. 8(A) illustrates normalized modulus error evolution for the eightreconstructed time states for noise free sufficiently oversampled data.In particular, FIG. 8(A) illustrates normalized modulus error as afunction of iteration number for w=0.

FIG. 8(B) illustrates normalized modulus error evolution for the eightreconstructed time states for noise free sufficiently oversampled data.In particular, FIG. 8(B) illustrates normalized modulus error as afunction of iteration number for w=0:5.

FIG. 9(A) illustrates cross sections of reconstructions from real datashowing the amplitude and phase for two time states with ½ the requiredoversampling for t=12 minutes. The images shown are cross sections ofthe states shown as isosurfaces in FIG. 6. Shown are w=0 and w=0.01. Thetrue solution, reconstructed with the full data set, is also shown.

FIG. 9(B) illustrates cross sections of reconstructions from real datashowing the amplitude and phase for two time states with ½ the requiredoversampling for t=42 minutes. The images shown are cross sections ofthe states shown as isosurfaces in FIG. 6. Shown are w=0 and w=0.01. Thetrue solution, reconstructed with the full data set, is also shown.

FIG. 10(A) illustrates the average (over all time states and randomstarts) normalized modulus error as a function of the scalar weight wfor ⅓, 1/20, and 1/84 of the required oversampling. The modulus errorreported here is computed with respect to the data with the requiredoversampling. Error bars represent the standard deviation of thedifferent average (over all time states) values obtained from 10different random starts. In particular, FIG. 10(A) shows results for atime sequence consisting of t=0; 18; 34, 50; 66; and 76 minutes.

FIG. 10(B) illustrates the average (over all time states and randomstarts) normalized modulus error as a function of the scalar weight wfor ⅓, 1/20, and 1/84 of the required oversampling. The modulus errorreported here is computed with respect to the data with the requiredoversampling. Error bars represent the standard deviation of thedifferent average (over all time states) values obtained from 10different random starts. In particular, FIG. 10(B) shows results for atime sequence consisting of t=0; 34; 66; and 76 minutes.

FIG. 10(C) illustrates the average (over all time states and randomstarts) normalized modulus error as a function of the scalar weight wfor ⅓, 1/20, and 1/84 of the required oversampling. The modulus errorreported here is computed with respect to the data with the requiredoversampling. Error bars represent the standard deviation of thedifferent average (over all time states) values obtained from 10different random starts. In particular, FIG. 10(C) shows results for atime sequence consisting of t=0; 34; and 76 minutes.

FIG. 11(A) illustrates the average (over all time states and randomstarts) normalized modulus error as a function of the scalar weight wfor ⅓, 1/20, and 1/84 of the required oversampling. The modulus errorreported here is computed with respect to the data with the requiredoversampling. Error bars represent the standard deviation of thedifferent average (over all time states) values obtained from 10different random starts. In particular, FIG. 11(A) shows results for atime sequence consisting of t=0, 6, 10, 14, 18, 22, 26, 30, 34, and 38minutes.

FIG. 11(B) illustrates the average (over all time states and randomstarts) normalized modulus error as a function of the scalar weight wfor ⅓, 1/20, and 1/84 of the required oversampling. The modulus errorreported here is computed with respect to the data with the requiredoversampling. Error bars represent the standard deviation of thedifferent average (over all time states) values obtained from 10different random starts. In particular, FIG. 11(B) shows results for atime sequence consisting of t=0, 2, 4, 6, 8, 10, 12, 14, 16, and 18minutes.

FIG. 12 is a flowchart illustrating the steps of a chrono CDI algorithm(method for phase retrieval) for sparse measurement sampling duringdynamic processes.

DETAILED DESCRIPTION

Before turning to the figures, which illustrate the exemplaryembodiments in detail, it should be understood that the presentapplication is not limited to the details or methodology set forth inthe description or illustrated in the figures. It should also beunderstood that the terminology is for the purpose of description onlyand should not be regarded as limiting.

Referring, in general, to the figures, a method for phase retrieval forsparse measurement sampling during dynamic processes is provided. Asused herein, “dynamic processes” refers to a process in which thestructure or position of an object to be imaged changes with time. Formost physical processes, structural evolution is a continuous processthat introduces structural redundancy when measured as a time series.The method described in the embodiments of the present applicationexploit this redundancy to allow for reduced oversampling (i.e., lessthan the conventionally required factor of 2), thereby improving themeasurement rate. This method (hereinafter referred to as “chrono CDI”)improves the temporal resolution of BCDI by reducing the oversamplingrequirement along one dimension (q_(z)) at a given time step withoutsignificantly compromising image fidelity. As used herein, imagefidelity means that the reconstructed image is still interpretable. Onemethod of quantitatively determining whether image fidelity ismaintained is to sum the errors (i.e., the difference between the realand reconstructed image) on a pixel by pixel basis between the realimage (i.e., an image based on a data set sampled at the Nyquistfrequency) and the reconstructed image, and determine whether the sumexceeds a predetermined level of acceptable error. This can be done inreal or reciprocal space. In one example, the predetermined level ofacceptable error is less than or equal to 10% per pixel.

in one embodiment (see FIG. 12), according to a process 100, a 3D dataset D_(t) is acquired for a single time state ρ_(t). In Step 110, acoherent beam illuminates an object to be imaged. In Step 111, the beamscattered by the object produces a first diffraction pattern DP₁ that ismeasured by an area detector. In Step 112, the object is rotated arounda tilt axis. In Step 113, the coherent beam illuminates the object (at adifferent position than Step 110 due to the rotation of the object). InStep 114, a second diffraction pattern DP₂ is measured by the areadetector. Steps 112-114 are repeated for a predetermined number of timessuch that a sequence of 2D diffraction patterns are measured atdifferent sample orientations. In other words, The sequence of 2Ddiffraction patterns (i.e., the 2D data collected from Steps 110, 111and all repetitions of Step 112-114) are stacked to form a 3D data setD_(t) measured at a single time state ρ_(t). In others, the 3D data setD_(t) includes each of the diffraction patterns DP₁, DP₂, DP₃ . . .DP_(n) collected at each position of the object during the time stateρ_(t). The time between the acquisition of each diffraction pattern DP₁,DP₂, DP₃ . . . DP_(n) is equal to the time it takes to reposition theobject (Step 112), re-illuminate the object with the coherent beam (Step113), and measure the resulting diffraction pattern (Step 114).

In the chrono CDI method, the process 100 is repeated a predeterminednumber of times in order to obtain a 3D data set for each of a pluralityof time states ρ₁, ρ₂, ρ₃, . . . ρ_(n). The number of time statesrequired depends on the dynamics of the process. Typically, the processneeds to be one order of magnitude slower than the time it takes toobtain a single 3D data set (i.e., one order of magnitude slower thanρ_(t)). To explain, a 3D data set D_(t1) is obtained for a first timestate ρ_(t), no measurements are acquired for an interval of timeT_(rest), a 3D data set D_(t2) is obtained for a second time state ρ₂,no measurements are acquired for the interval of time T_(rest), and theprocess is repeated until a 3D data D_(tn) is obtained for a time stateρ_(n). The interval of time T_(rest) between each time state is selectedbased on how rapidly the structure of the object is changing in thedynamic process. In particular, the faster the structure of the objectis changing, the smaller the time T_(rest). The slower the structure ofthe object is changing, the larger the time T_(rest).

in one example, the 2D diffraction patterns that comprise the 3D dataset D_(t) are collected continuously with the time between acquiringdifferent measurements in the 3D data. set being about 10-20 seconds.The time it takes to acquire the entire 3D data (i.e., the duration of asingle time state ρ_(t)) may be, for example, 6 minutes. The timeT_(rest) between each time state ρ_(t) may be, for example, the time ittakes to reposition the object to the position in which the object wasprovided when the first diffraction pattern DP₁ was measured in thefirst time state ρ₁ (e.g., 15-30 seconds) or longer.

After a 3D data set D_(t) is acquired for each time state ρ_(t), thedata is processed by a computer programmed to reconstruct a 3D image ofthe object that shows the structure of the object at any given timestate ρ_(t).

In one example, Steps 110-114 are steps performed in BCDI. FIG. 1(A)shows a schematic of a Bragg rocking curve. In Step 110, a coherentx-ray beam k_(i) illuminates an object to be imaged (e.g., ananocrystal) and in Step 111, the beam scattered by the object producesa first diffraction pattern (e.g., in an area detector plane 1) that ismeasured by an area detector. Creating the rocking curve entailsrotating the sample with respect to the incident x-ray beam illuminatingthe object with the coherent x-ray beam and measuring a subsequentdiffraction pattern (e.g., in an area detector plane 2) using the areadetector (Steps 112-114). The sample rotation, labeled schematically byθ_(start) and θ_(end), displaces the scattering vector q=k_(f)-k_(i)from the reciprocal-space lattice point G_(hkl), the Bragg reflectioncondition for the HKL lattice planes, so that the 3D intensitydistribution can be appropriately sampled and the structure of thenanocrystal can be reconstructed. The series of 2D measurements fromeach of the area detector planes 1-7 are stacked to form a 3D data setD_(t), where t represents a time index in a series of sequentialrocking-curve measurements. As discussed above, the 3D data set D_(t)includes each of the diffraction patterns DP₁, DP₂, DP₃ . . . DP_(n)collected at each position of the object (i.e., each of the areadetector planes 1-7) during a single time state ρ_(t). In the chrono CDImethod of the present application, the number of diffraction patternsDP₁, D₂, DP₃, . . . DP_(n) that comprise the 3D data set D_(t) for eachtime state ρ_(t) can be reduced while still maintaining the fidelity ofthe reconstructed 3D image of the object.

Conventional phase retrieval algorithms require an oversampling of atleast two of the diffraction patterns in all three dimensions. Inaddition, the nanocrystal must be approximately static over themeasurement time (i.e., time state) ρ_(t) while D_(t) is collected,which limits the dynamic time scale that can be observed. Each timestate ρ_(t) is considered independently. Due to these requirements all2D measurements (i.e., in area detector planes 1-7 in FIG. 1(A)) must betaken. Reduction in the number of measurements (i.e., omission of thedata from any one of the detector planes 1-7) would result in a blurryimage. As discussed above, for conventional phase retrieval algorithms,the pattern must be sampled at twice the Nyquist frequency to satisfythe oversampling requirements. For example, for 100-500 nm nanocrystals,approximately 60-80 measurements (slices) are required.

In contrast, in chrono CDI, to improve the time resolution, only some 2Dmeasurements (for example, every other diffraction pattern DP₁, DP₂, DP₃. . . DP_(n) as indicated by the asterisks in FIG. 1(A)) could be taken.The number of measurements (i.e., diffraction patterns) measured pereach time state ρ_(t) in chrono CDI is less than a number ofmeasurements (i.e., diffraction patterns) measured in conventional phaseretrieval algorithms without comprising the fidelity of thereconstructed 3D image. The reduction in the number of measurementsrequired in chrono CDI is possible because information from each timestate ρ₁, ρ₂, ρ₃, . . . . ρ_(n) is used in the reconstruction of a 3Dimage for any given time state ρ_(t). In other words, in chrono CDI, thereconstruction of the 3D image is dependent on the diffraction patternsobtained in every time state ρ_(t), unlike the conventional phaseretrieval algorithms which consider each time state ρ_(t) independently.In other words, in chrono CDI, the number of measurements taken whilestill maintaining image fidelity can be reduced to a number below theoversampling requirement (i.e., a number less than the Nyquistfrequency). The faster the dynamic process, the fewer number ofmeasurements can be eliminated, and the slower the dynamic process, thegreater number of measurements can be eliminated. This is due to thenearest neighbor correlation (described below).

To illustrate an example of structure of the object changing in thedynamic process, FIG. 1(B) shows a reconstructed image from dataacquired at three selected time states during which a single crystalpalladium (Pd) nanocube (85 nm side length) is exposed to hydrogen gas.Palladium nanocubes were synthesized by using a wet chemistry method andthen spin casted onto a silicon substrate for the measurement. Theaverage nanocube side length is 85 nm. A double crystal monochromatorwas used to select x-rays with a photon energy of 8.919 keV with 1 eVbandwidth and longitudinal coherence length of 0.7 μm. A set ofKirkpatrick Baez mirrors was used to focus the beam to approximately1.5×1.5 μm². The silicon substrate was placed into a gas flow cell. Thedetector was set to a (111) Pd Bragg peak angle and the sample wasscanned in the x-ray beam until a (111) Pd diffraction peak illuminatedthe detector. The particles are randomly oriented and thereforetypically at most one diffraction pattern illuminates the area detector.

The rocking curve around the (111) Bragg reflection was collected byrecording 2D coherent diffraction patterns with an x-ray sensitive areadetector (Medipix 3, 55 μm pixel size) placed at 0.26 m away from thesample around an angle of 2Θ=36 deg. An angular range of (ΔΘ=±0:15 deg)was sampled at 51 equally spaced intervals. The total time for each 3Dmeasurement was approximately 2 minutes. Two to three full 3D datasetswere taken for each palladium nanocrystal in a helium environment. The3.6% mole fraction H₂ gas in He was then flowed to increase the partialpressure of H₂ above zero. A Mylar window on the gas flow cell allowsfor a pressure slightly greater than atmospheric pressure. The pressurewas left constant while 3D datasets were continuously collected atapproximately 2-minute intervals.

The absolute value of the image corresponds to the Bragg-diffractingelectron density, while the phase φ of the image is proportional to acomponent of the vector displacement field u via φ=u·Q. In this case,the Pd (111) Bragg peak was measured and φ˜u₁₁₁. In the Pd nanocube,hydrogen intercalation initially causes displacement field changes (t=42min) before morphological changes occur (t=76 min) due to the hydridingphase transformation. The time evolution of the nanocube structure shownin FIG. 1(B) was determined from BCDI experiments performed with anoversampling of 3 in q_(z) at Sector 34-ID-C of the Advanced PhotonSource at Argonne National Laboratory. Each complete measurement tookapproximately 2 minutes.

After the data is acquired, the data is processed to reconstruct a 3Dimage. In order to recover the phase information, Fourier-basediterative phase retrieval algorithms are utilized. In conventional phaseretrieval algorithms, the disagreement or error between measurement andimage Fourier transform is minimized. In particular, the functionminimized by the error reduction phase retrieval algorithms is themodulus error ρ

, which measures the agreement between the reconstruction's Fouriermoduli and the measured moduli,

$\left. {{ɛ_{\mathcal{M}}^{2}\left( {\rho,D} \right)} = {\sum\limits_{q}^{\;}{{{\overset{\sim}{\rho}} - \sqrt{D}}}^{2}}}, \right|$

where ρ is the 3D reconstructed object (in real space), D is the 3Dfar-field intensity measurement, q is the reciprocal-space coordinate,ρ^(˜)=

[ρ], and

is the Fourier transform. Different choices of the function to beminimized lead to different phase retrieval algorithms. Conventionalphase retrieval algorithms utilize data from a single time state ., datacollected in the measurement time ρ_(t)).

As discussed above, in the chrono CDI method of the present application,a plurality of 3D data sets D_(t) are acquired, one for each of aplurality of time states ρ_(t). The entire measurement time series(every data set D_(t) from every time state ρ_(t)), which is typically acontinuous physical process, is incorporated into a chrono CDI phaseretrieval algorithm that allows the oversampling requirement at eachtime step to be reduced. The chrono CDI algorithm includes amiscorrelation term that depends on reconstructions at other timestates, described according to Equation 1 below:

$\sum\limits_{t}^{\;}{\left\lbrack {{ɛ_{\mathcal{M}}^{2}\left( {\rho_{t},D_{t}} \right)} + {\sum\limits_{t^{\prime} \neq t}^{\;}{{w\left( {t,t^{\prime}} \right)}{\mu \left( {\rho_{t},\rho_{t^{\prime}}} \right)}}}} \right\rbrack.}$

In this expression, t indexes the time states, w(t, t′) is the weight,t≠t′, and μ(ρt, ρt′) is the miscorrelation term. In Equation 1, thefirst term in the bracket measures the agreement between the measureddata and the reconstructed image The second term in the bracketdescribes how similar the time states are to one another. The secondterm in the bracket represents the correlation between the time states.The nearest-neighbor correlations in time, is considered to be a scalarweight parameter w≥0, and a functional form for the miscorrelation of

${\mu \left( {\rho_{t},\rho_{t^{\prime}}} \right)} = {{\sum\limits_{r}^{\;}{{\rho_{t} - \rho_{t - 1}}}^{2}} + {\sum\limits_{r}^{\;}{{{\rho_{t} - \rho_{t + 1}}}^{2}.}}}$

The tth term of Equation 1 then becomes:

$\begin{matrix}{{ɛ_{\mathcal{M}}^{2}\left( {\rho_{t},D_{t}} \right)} + {{w\left( {{\sum\limits_{r}^{\;}{{\rho_{t} - \rho_{t - 1}}}^{2}} + {\sum\limits_{r}^{\;}{{\rho_{t} - \rho_{t + 1}}}^{2}}} \right)}.}} & (2)\end{matrix}$

Although other forms are possible, this form has the advantage of beingcomputationally inexpensive. The iterative algorithm is derived byminimizing Equation 2 summed over t.

An example of the algorithm for both the error reduction like and hybridinput output like update steps can be seen in the Matlab pseudocodebelow:

if ERflag=1

% error reduction steprho(:,:,:,qq)=(1/(1+2*weights))*support(:,:,:,qq).*(Pmrho(:,:,:,qq)+. ..(weights)*(rho(:,:,:,qq+1)+rho(:,:,:,qq−1)));else

%HIO:

cterm=2*rho(:,:,:,qq)−rho(:,:,:,qq+1)−rho(:,:,:,qq−1);sc=1−support(:,:,:,qq);rho(:,:,:,qq)=support(:,:,:,qq).*Pmrho(:,:,:,qq)+. . .sc*(rho(:,:,:,qq)−beta*Pmrho(:,:,:,qq))−. . .weights*support(:,:,:,qq).*cterm;end

where qq indexes the time variable, rho is the current best guess forthe reconstruction, support is the current support used,Pmrho(:,:,:,qq)=(ifftn(fftshift(Psi mod(:,:,:,qq)))), with Psimod(:,:;:,qq)=data(:,:,:,qq).*exp(li*angle(Psi2(:,:,:,qq))); andPsi2(:,:,:,qq)=fftshift(fftn((rho(:,:,:,qq))));

To evaluation the performance of the chrono CDI algorithm, iterativephase retrieval was carried out on noise-free, simulated data, as wellas on measured experimental data with different amounts of oversampling.

Simulated Data With Required Oversampling

In this case, the simulated data D_(t) ^(sim) are generated by 3DFourier transforms of each complex valued Pd nanocube reconstruction inthe time series after zero padding to meet the oversampling (OS)requirements of phase retrieval. As discussed above, as used in thisapplication, an oversampling of 2 is used as the required oversampling.In practice, this means the cube size was half of the array size in allthree dimensions. The time sequence (i.e., different time states)considered consists of t=0, 12, 18, 26, 34, 42, 50, 58, 66, and 76minutes. This time sequence is approximately equally spaced and, asshown in FIG. 2, has varying amounts of nearest-neighbor correlation,with an average nearest-neighbor correlation coefficient of 61%. Thecorrelation coefficient c(t,t′)ϵ[−1, 1] is defined between two 3Ddisplacement fields at time states t and t′ by

$\frac{\sum\limits_{r}^{\;}{\left\lbrack {{u_{111}\left( {r,t} \right)} - {{\overset{\_}{u}}_{111}\left( {r,t} \right)}} \right\rbrack \left\lbrack {{u_{111}\left( {r,t^{\prime}} \right)} - {{\overset{\_}{u}}_{111}\left( {r,t^{\prime}} \right)}} \right\rbrack}}{\sqrt{\sum\limits_{r}^{\;}\left\lbrack {{u_{111}\left( {r,t} \right)} - {{\overset{\_}{u}}_{111}\left( {r,t} \right)}} \right\rbrack^{2}}\sqrt{\sum\limits_{r}^{\;}\left\lbrack {{u_{111}\left( {r,t^{\prime}} \right)} - {{\overset{\_}{u}}_{111}\left( {r,t^{\prime}} \right)}} \right\rbrack^{2}}},$

where u₁₁₁ is the displacement field projection and ū₁₁₁ is the averagedisplacement field over the particle.

FIG. 3(A) shows the correlation coefficient matrix for the chosen timesequence. FIG. 2 is a plot of the superdiagonal matrix values. Thechosen time sequence is a good balance between having a smooth evolutionbetween nearest neighbors and having a large change over the whole timesequence (both the displacement and the amplitude change significantly).The first numerical test of chrono CDT reconstructs the sequence ρ_(t)from the sequence of D_(t) ^(sim) with the required oversampling.

The algorithm uses random initial starts and alternates between theerror reduction (ER) and the hybrid input-output (HIO) algorithms usinga feedback parameter of β=0.7; the support is fixed to the size of theobject and is not evolved during the iterative process. At iterationnumbers N=100n, for n=1, 2, . . . , 18, the algorithm tests whether allreconstructions are correctly oriented with respect to the known initial(t=0 min) and final (t=76 min) states by testing whether they areconjugated and reflected (“twin”) solutions. In order to have a knowninitial state and final state, the initial and final states are measuredwith the traditional number of measurements that satisfy theoversampling requirements (i.e., at the Nyquist frequency).

Although reconstructing the “twin” image does not affect ϵ

, it will negatively impact μ, resulting in an artificially high totalobjective. One constraint used is that the ρ_(t) of the initial andfinal states are known in real and diffraction space. This constraintcan be achieved by measuring diffraction data sets at the requiredoversampling before the experimental dynamics start and after nosignificant changes are seen in the diffraction data.

FIG. 3(B) shows the errors ε

(ρ_(t), D_(t) ^(sim)) and μ(ρ_(t), ρ_(t)′), averaged over allreconstructed time states and over ten random starts, as a function ofthe scalar weight w. Both errors are normalized by the total intensityin the image. FIGS. 8(A) and 8(B) show the modulus error ε

as a function of iteration number for two scalar weight values for eightreconstructed time states for noise free sufficiently oversampled date.In particular, FIG. 8(A) illustrates the normalized modulus error as afunction of iteration number for w 32 0, while FIG. 8(B) illustrates thenormalized modulus error as a function of iteration number for w=0.5.The initial and final states (t=0 and t=76 min, respectively) are knownin real space. When w=0, the modulus error is the lowest, and themiscorrelation term is the largest. These results are expected becausethe data are noise free and oversampled at the required oversamplingsuch that a unique solution is fully determined for each D_(t) ^(sim).The weight w=0 corresponds to the case when no correlations are takeninto account. As w increases, μ(ρ_(t), ρ_(t)′) decreases, and ε

increases for the solution set {ρ_(t)} because their relativecontributions to the total objective change. With data sampled at therequired oversampling, including information from neighboring time stepsin a time series will not improve each individual reconstruction becausethe complete 3D structural description of the sample at each time isuniquely encoded in the 3D coherent intensity pattern.

Simulated Data With Reduced Oversampling

Next, the inventors explored how the additional redundancy from the timeseries can compensate for reduced oversampling during the rocking curve(e.g., oversampling at less than a factor of 2) at a given time step byreconstructing the time series D_(t) ^(sim) discussed previously, butwith different degrees of reduced sampling in q_(z). To start, everythird 2D diffraction measurement of the original 84 2D diffractionmeasurements was selected to form D_(t) ^(sim) for all times except theinitial and final time. This leads to data that have ⅓ of the requiredoversampling. If such a time series were measured experimentally, themeasurement time would be reduced by a factor of 3. in assessingalgorithm performance, the modulus error was calculated by comparing thefar-field exit wave of the reconstructions with the data sampled at therequired oversampling. As before, the initial and final states areknown, the support is known, and alternating ER/HIO is used as describedpreviously.

FIG. 4(A) shows the average (over all time states and random starts)modulus error as a function of the scalar weight w for varying amountsof oversampling. The average normalized modulus error does not changefrom w=0 to w=10⁻⁴. Unlike the results using the required. oversampling(see FIG. 3(B)) where the lowest modulus error occurs for w=0 (no timecorrelation), in the cases where q_(z) has been sampled at ⅓ (bottomline of FIG. 4(A)) and 1/20 (middle line of FIG. 4(A)) of the requiredoversampling, a minimum in the modulus error is observed at a value ofw=0.01. This modulus error is computed with respect to the data setsthat are sampled at the required oversampling, and thus thereconstructions at these minima are the “best” solutions. When only 1/84(top line of FIG. 4(A)) of the required oversampling is used (i.e., asingle slice from the rocking curve), the modulus error decreases withincreasing w and approaches a constant value, which is near thenormalized average modulus error when all the reconstructed time statesare set to the average of the initial and final state (black dashedhorizontal line). In this case, simply using an average of the initialand final states outperforms the reconstruction algorithm, indicatingthat there are insufficient reciprocal space constraints and that theoversampling in q_(z) is too low.

On average, w=0.01 improves the reconstructed time sequence relative tow=0, which takes no time correlation into account, for up to 1/20 of therequired oversampling. However, it is not clear from the plot of theaverage whether all time states are being improved equally. FIG. 4(B)shows the normalized modulus error at each time state for w=0, 0.01, and0.5 at ⅓ of the required oversampling for a particular random start. Thestates nearest to the known states (t=0 and t=76 min) have lower moduluserrors, as expected. By comparing w=0 with w=0.01, we see that theimprovement occurs in all the intermediate states except the state att=66 min, which remains essentially unchanged. These results demonstratethat the algorithm improves all reconstructions, even those leastcorrelated with their neighbors (see FIG. 2 for a plot of thenearest-neighbor correlation). The benefits of chrono CDI are clear whenthe required oversampling is reduced by up to a factor of approximately1/20. In these cases, enforcing a degree of nearest-time-step,real-space correlation provides an additional constraint that improvesthe reconstruction at all intermediate times relative to what can beachieved using conventional phase retrieval. Finally, all of thereconstructions on simulated data were rerun with an unknown supportthat was updated via the “shrink-wrap” algorithm to make a bettercomparison with the experimental data. The conclusions discussedpreviously remain valid.

Experimental Date

Next, chrono CDI was demonstrated on experimental rocking curve data. Tosimulate varying degrees of reduced oversampling, a subset of theoriginal 2D measurements was selected from the experimental data sets.FIG. 5 shows example 2D experimental diffraction measurements from thePd (111) Bragg rocking curve. An oversampling of approximately 3 in qzwas used during the original measurement. The reconstruction algorithmis the same as described previously, except that the support is notknown a priori. Instead, an initial box half the array size in eachdimension is used, and the support is updated with the shrink-wrapalgorithm using a Gaussian function with a threshold of 0.01 andstandard deviation of 1.

FIGS. 6(A) and 6(B) show chrono CDI reconstructions for tworepresentative time states of experimental diffraction data from Pdnanocubes undergoing structural transformations when exposed to hydrogengas. As before, different amounts of oversampling were investigated. Theisosurfaces shown correspond to the reconstructed Bragg electrondensity, while the color map corresponds to the image phase φ, which isproportional to the u111 displacement field. FIG. 6(A) showsreconstructions when ½ of the required oversampling in q_(z) is used.Every third slice of the original data (oversampled at a factor of 3 inq_(z)) was used to generate data. This corresponds to an oversampling of1, which is ½ the required oversampling of 2. The reconstruction forw=0.01 is much improved compared with w=0 and is similar in morphologyand lattice displacement to the true solution. The true solution isreconstructed using the complete data set. FIG. 6(B) shows that the sameconclusion holds for t=42 min. FIG. 9 illustrates central cross sectionsthat show the amplitude and phase distributions inside the crystal. Theaverage normalized modulus error of the time sequence is improved from0.2 to 0.1 by including nearest-neighbor information (via w=0.01).Although the reconstructions do not match exactly, the results conveythe same overall physical changes in the crystal.

FIG. 7 shows that when 3/10 of the required oversampling is used(corresponding to every fifth slice of the original data), majordifferences in both the reconstructed Bragg electron density anddisplacement fields arise as compared with the full-rocking-curvereconstructions. There are also disagreements in the reconstructedphases (proportional to the displacements). Thus, the chrono CDI appliedto this particular set of measurement data for the chosen time sequencecould have decreased the measurement time by a factor of 2 withoutlosing the essential physics of the transforming crystal. For simulateddata it could have reduced the time by up to a factor of 20. Thediscrepancy is due to the finite scattering extent in reciprocal spaceof the real data in addition to noise.

The success of the algorithm is directly tied to how correlated theintermediate time states are among themselves and to the initial andfinal states. This correlation can be tuned by changing the time scaleof the dynamics in the experiment or by taking finer time steps. Morecorrelated time sequences will lead to larger optimal w values and tobetter improvement in temporal resolution, while less correlated timeseries will lead to smaller optimal w values and to less improvement intemporal resolution.

Additional time sequences with different numbers of intermediate timeare illustrated in FIGS. 10(A)-10(C). In particular, FIGS. 10(A)-10(C)illustrate the average (over all time states and random starts)normalized modulus error as a function of the scalar weight w for ⅓,1/20, and 1/84 of the required oversampling. w=0 (not shown) producesthe same average normalized modulus error values as w=10⁻⁴. The moduluserror reported here is computed with respect to the data with therequired oversampling. Error bars represent the standard deviation ofthe different average (over all time states) values obtained from 10different random starts. FIG. 10(A) shows results for a time sequenceconsisting of t=0, 18, 34, 50, 66, and 76 minutes. FIG. 10(B) showresults for a time sequence consisting of t=0, 34, 66, and 76 minutes.FIG. 10(C) shows results for a time sequence consisting of t=0, 34, 76minutes, FIGS. 10(A)-10(C) demonstrate that w=0.01 is fairly robust whenless intermediate time states are used.

FIGS. 11(A) and 11(B) illustrate the average (over all time states andrandom starts) normalized modulus error as a function of the scalarweight w for ⅓, 1/20, and 1/84 of the required oversampling. w=0 (notshown) produces the same average normalized modulus error values asw=10⁻⁴. The modulus error reported here is computed with respect to thedata with the required oversampling. Error bars represent the standarddeviation of the different average (over all time states) valuesobtained from 10 different random starts. FIG. 11(A) shows results for atime sequence consisting of t=0, 6, 10, 14, 18, 22, 26, 30, 34, and 38minutes. FIG. 11(B) shows results for a time sequence consisting of t=0,2, 4, 6, 8, 10, 12, 14, 16, and 18 minutes. FIGS. 11(A) and 11(B) showthat w=0.01 is fairly robust when different time sequences are used.

As discussed above, by incorporating the entire measurement time series,which is typically a continuous physical process, into phase retrievalallows the oversampling requirement at each time step to be reduced,leading to a subsequent improvement in the temporal resolution by afactor of 2-20 times. By incorporating the redundancy from the entiretime series into each individual measurement, less time can be takenduring each measurement and the time resolution is increased. Theincreased time resolution will allow imaging of faster dynamics and ofradiation-dose-sensitive samples. Although use of the “chrono CDI”approach was discussed in conjunction with BCDI, the chrono CDI approachmay find use in improving the time resolution in other imagingtechniques.

As demonstrated above, the chrono CDI phase retrieval algorithm improvesthe time resolution of BCDI by a factor of 2 for experimental data and20 for simulated data. The algorithm thereby enables BCDI investigationsof faster processes and can be used to limit radiation dose. The timeresolution improvements we demonstrate are achieved by reducing thenumber of 2D measurements made during a 3D Bragg rocking curve, leadingto data sets with less than the required oversampling in qz at eachintermediate time step. The rocking curves across the entire time seriesare reconstructed simultaneously, enforcing a degree of real-spacecorrelation between solutions at neighboring time steps to account forthe reduced oversampling of each individual measurement. In otherembodiments, the algorithm and its variations may be useful forimproving the time resolution of other imaging techniques, such asptychography (ptychographic coherent imaging), tomography (e.g., CTscans), etc., where there is a continuous relationship in real spacebetween nearest-neighbor time states.

The construction and arrangements of the method for phase retrieval, asshown in the various exemplary embodiments, are illustrative only.Although only a few embodiments have been described in detail in thisdisclosure, many modifications are possible (e.g., variations in sizes,dimensions, structures, shapes and proportions of the various elements,values of parameters, mounting arrangements, use of materials, colors,orientations, image processing and segmentation algorithms, etc. withoutmaterially departing from the novel teachings and advantages of thesubject matter described herein. Some elements shown as integrallyformed may be constructed of multiple parts or elements, the position ofelements may be reversed or otherwise varied, and the nature or numberof discrete elements or positions may be altered or varied. The order orsequence of any process, logical algorithm, or method steps may bevaried or re-sequenced according to alternative embodiments. Othersubstitutions, modifications, changes and omissions may also be made inthe design, operating conditions and arrangement of the variousexemplary embodiments without departing from the scope of the presentinvention.

As utilized herein, the terms “approximately,” “about,” “substantially”,and similar terms are intended to have a broad meaning in harmony withthe common and accepted usage by those of ordinary skill in the art towhich the subject matter of this disclosure pertains. It should beunderstood by those of skill in the art who review this disclosure thatthese terms are intended to allow a description of certain featuresdescribed and claimed without restricting the scope of these features tothe precise numerical ranges provided. Accordingly, these terms shouldbe interpreted as indicating that insubstantial or inconsequentialmodifications or alterations of the subject matter described and claimedare considered to be within the scope of the invention as recited in theappended claims.

References herein to the positions of elements (e.g., “top,” “bottom,”“above,” “below,” etc.) are merely used to describe the orientation ofvarious elements in the FIGURES. It should be noted that the orientationof various elements may differ according to other exemplary embodiments,and that such variations are intended to be encompassed by the presentdisclosure.

With respect to the use of substantially any plural and/or singularterms herein, those having skill in the art can translate from theplural to the singular and/or from the singular to the plural as isappropriate to the context and/or application. The varioussingular/plural permutations may be expressly set forth herein for thesake of clarity.

The chrono CDI algorithm/method described above may be executed by acomputer programmed to perform the steps of the algorithm. Embodimentsof the subject matter and the operations described in this specificationcan be implemented in digital electronic circuitry, or in computersoftware embodied on a tangible medium, firmware, or hardware, includingthe structures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. Embodiments ofthe subject matter described in this specification can be implemented asone or more computer programs, i.e., one or more modules of computerprogram instructions, encoded on one or more computer storage medium forexecution by, or to control the operation of, data processing apparatus.Alternatively or in addition, the program instructions can be encoded onan artificially-generated propagated signal, e.g., a machine-generatedelectrical, optical, or electromagnetic signal that is generated toencode information for transmission to suitable receiver apparatus forexecution by a data processing apparatus. A computer storage medium canbe, or be included in, a computer-readable storage device, acomputer-readable storage substrate, a random or serial access memoryarray or device, or a combination of one or more of them. Moreover,while a computer storage medium is not a propagated signal, a computerstorage medium can be a source or destination of computer programinstructions encoded in an artificially-generated propagated signal. Thecomputer storage medium can also be, or be included in, one or moreseparate components or media (e.g., multiple CDs, disks, or otherstorage devices). Accordingly, the computer storage medium may betangible and non-transitory.

The operations described in this specification can be implemented asoperations performed by a data processing apparatus or processingcircuit on data stored on one or more computer-readable storage devicesor received from other sources.

The apparatus can include special purpose logic circuitry, e.g., an FPGA(field programmable gate array) or an ASIC (application-specificintegrated circuit). The apparatus can also include, in addition tohardware, code that creates an execution environment for the computerprogram in question, e.g., code that constitutes processor firmware, aprotocol stack, a database management system, an operating system, across-platform runtime environment, a virtual machine, or a combinationof one or more of them. The apparatus and execution environment canrealize various different computing model infrastructures, such as webservices, distributed computing and grid computing infrastructures.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, object, orother unit suitable for use in a computing environment. A computerprogram may, but need not, correspond to a file in a file system. Aprogram can be stored in a portion of a file that holds other programsor data (e.g., one or more scripts stored in a markup languagedocument), in a single file dedicated to the program in question, or inmultiple coordinated files (e.g., files that store one or more modules,sub-programs, or portions of code). A computer program can be deployedto be executed on one computer or on multiple computers that are locatedat one site or distributed across multiple sites and interconnected by acommunication network.

The processes and logic flows described in this specification can beperformed by one or more programmable processors or processing circuitsexecuting one or more computer programs to perform actions by operatingon input data and generating output. The processes and logic flows canalso be performed by, and apparatus can also be implemented as, specialpurpose logic circuitry, e.g., an FPGA or an ASIC.

Processors or processing circuits suitable for the execution of acomputer program include, by way of example, both general and specialpurpose microprocessors, and any one or more processors of any kind ofdigital computer. Generally, a processor will receive instructions anddata from a read-only memory or a random access memory or both. Theessential elements of a computer are a processor for performing actionsin accordance with instructions and one or more memory devices forstoring instructions and data. Generally, a computer will also include,or be operatively coupled to receive data from or transfer data to, orboth, one or more mass storage devices for storing data, e.g., magnetic,magneto-optical disks, or optical disks. However, a computer need nothave such devices. Moreover, a computer can be embedded in anotherdevice, e.g., a mobile telephone, a personal digital assistant (PDA), amobile audio or video player, a game console, a Global PositioningSystem (GPS) receiver, or a portable storage device (e.g., a universalserial bus (USB) flash drive), to name just a few. Devices suitable forstoring computer program instructions and data include all forms ofnon-volatile memory, media and memory devices, including by way ofexample semiconductor memory devices, e.g., EPROM, EEPROM, and flashmemory devices; magnetic disks, e.g., internal hard disks or removabledisks; magneto-optical disks; and CD-ROM and DVD-ROM disks. Theprocessor and the memory can be supplemented by, or incorporated in,special purpose logic circuitry.

To provide for interaction with a user, embodiments of the subjectmatter described in this specification can be implemented on a computerhaving a display device, e.g., a CRT (cathode ray tube) or LCD (liquidcrystal display), OLED (organic light emitting diode), TFT (thin-filmtransistor), plasma, other flexible configuration, or any other monitorfor displaying information to the user and a keyboard, a pointingdevice, e.g., a mouse trackball, etc., or a touch screen, touch pad,etc., by which the user can provide input to the computer. Other kindsof devices can be used to provide for interaction with a user as well;for example, feedback provided to the user can be any form of sensoryfeedback, e.g., visual feedback, auditory feedback, or tactile feedback;and input from the user can be received in any form, including acoustic,speech, or tactile input. In addition, a computer can interact with auser by sending documents to and receiving documents from a device thatis used by the user; for example, by sending web pages to a web browseron a user's client device in response to requests received from the webbrowser.

What is claimed: 1) A method for retrieving phase information in acoherent diffraction imaging process, the method comprising: acquiring aplurality of 3D data sets, each 3D data set corresponding to one of aplurality of time states, where each 3D data set is acquired using thefollowing steps: illuminating an object to be imaged with a coherentbeam, the object positioned in a first position; measuring a first 2Ddiffraction pattern using an area detector that detects a number ofphotons or electrons in a beam scattered by the object; rotating theobject around a tilt axis thereof to a second position that is differentfrom the first position; re-illuminating the object with the coherentbeam, the object positioned in the second position; re-measuring asecond 2D diffraction pattern using the area detector; and repeating therotating, re-illuminating and re-measuring steps a predetermined numberof times such that each 3D data set includes a predetermined number ofdiffraction patterns; and reconstructing a 3D image of the object at agiven time state using information from the 3D data set from the giventime state and all of the other time states. 2) The method according toclaim 1, wherein the predetermined number of diffraction patterns in atleast one 3D data set is less than a minimum number of 2D diffractionpatterns required based on the Nyquist frequency. 3) The methodaccording to claim 2, wherein the 3D image reconstructed based on the atleast one 3D data set has image fidelity, where image fidelity isestablished when a sum of a difference between a real image and the 3Dimage reconstructed based on the at least one 3D data set on a pixel bypixel basis is less than or equal to a predetermined level of acceptableerror. 4) The method according to claim 3, wherein the predeterminedlevel of acceptable error is less than or equal to 10% per pixel. 5) Themethod according to claim 1, wherein the predetermined number ofdiffraction patterns in at least one 3D data set is ⅓ of a minimumnumber of 2D diffraction patterns required based on the Nyquistfrequency. 6) The method according to claim 1, wherein the plurality of3D data sets include a 3D data set acquired at an initial time state, a3D data set acquired at a final time state, and at least one 3D data setacquired at an intermediate time state between the initial time stateand the final time state. 7) The method according to claim 6, whereinthe predetermined number of diffraction patterns in the initial timestate is equal to a minimum number of 2D diffraction patterns requiredbased on the Nyquist frequency, wherein the predetermined number ofdiffraction patterns in the final time state is equal to the minimumnumber of 2D diffraction patterns required based on the Nyquistfrequency, and wherein the predetermined number of diffraction patternsin the intermediate time state is less than to the minimum number of 2Ddiffraction patterns required based on the Nyquist frequency. 8) Themethod according to claim 1, wherein the plurality of 3D data setsinclude a 3D data set acquired at an initial time state, a 3D data setacquired at a final time state, and a plurality of 3D data sets acquiredat intermediate time states between the initial time state and the finaltime state. 9) The method according to claim 1, wherein the step ofreconstructing the 3D image of the object is repeated for each of theplurality of time states to produce a series of 3D images that show achange in a structure of the object over time during a dynamic process.10) The method according to claim 9, wherein structural changes thatoccur during the dynamic process occur at a rate that is one magnitudeslower than a duration of each time state. 11) The method according toclaim 1, wherein the object is a nanocrystal having a size of 100-500nm, and wherein the step of reconstructing the 3D image of thenanocrystal is repeated for each of the plurality of time states toproduce a series of 3D images that show a change in a structure of thenanocrystal over time. 12) The method according to claim 11, wherein thepredetermined number of diffraction patterns in each of the 3D data setsis 60 to
 80. 13. The method according to claim 1, wherein a duration ofeach time state is less than a duration of time needed to obtain aminimum number of 2D diffraction patterns required based on the Nyquistfrequency.